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is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Damage in peridynamics formulations</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Damage in peridynamics formulations</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/kaipartmann/Peridynamics.jl/blob/main/docs/src/expl_damage.md#" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="expl_dmg"><a class="docs-heading-anchor" href="#expl_dmg">Damage in peridynamics formulations</a><a id="expl_dmg-1"></a><a class="docs-heading-anchor-permalink" href="#expl_dmg" title="Permalink"></a></h1><h2 id="Bond-based-formulation"><a class="docs-heading-anchor" href="#Bond-based-formulation">Bond-based formulation</a><a id="Bond-based-formulation-1"></a><a class="docs-heading-anchor-permalink" href="#Bond-based-formulation" title="Permalink"></a></h2><p class="math-container">\[ \boldsymbol{f} = d^{ij} \, c \, \varepsilon^{ij} \, \boldsymbol{n} \; .\]</p><p>with</p><table><tr><th style="text-align: left">size</th><th style="text-align: left">symbol</th><th style="text-align: left">unit</th></tr><tr><td style="text-align: left">bond failure</td><td style="text-align: left"><span>$d^{ij} \in \{0;1\}$</span></td><td style="text-align: left"><span>$[-]$</span></td></tr></table><p>In this expression, bond failure <span>$d^{ij}$</span> represents whether the bond between points <span>$i$</span> and <span>$j$</span> is intact (<span>$d^{ij}=1$</span>) or damaged (<span>$d^{ij}=0$</span>).</p><h2 id="Ordinary-state-based-formulation"><a class="docs-heading-anchor" href="#Ordinary-state-based-formulation">Ordinary state-based formulation</a><a id="Ordinary-state-based-formulation-1"></a><a class="docs-heading-anchor-permalink" href="#Ordinary-state-based-formulation" title="Permalink"></a></h2><p>TODO</p><h2 id="Non-ordinary-state-based-formulation"><a class="docs-heading-anchor" href="#Non-ordinary-state-based-formulation">Non-ordinary state-based formulation</a><a id="Non-ordinary-state-based-formulation-1"></a><a class="docs-heading-anchor-permalink" href="#Non-ordinary-state-based-formulation" title="Permalink"></a></h2><p>TODO</p><h2 id="Continuum-kinematics-inspired-formulation"><a class="docs-heading-anchor" href="#Continuum-kinematics-inspired-formulation">Continuum-kinematics-inspired formulation</a><a id="Continuum-kinematics-inspired-formulation-1"></a><a class="docs-heading-anchor-permalink" href="#Continuum-kinematics-inspired-formulation" title="Permalink"></a></h2><p class="math-container">\[    \boldsymbol{b}_1^{{\mathrm{int}},i} = \int_{\mathcal{H}_1^i} d^{ij} C_1 \left( \frac{l^{ij}}{L^{ij}} - 1 \right) \frac{\boldsymbol{\Delta x}^{ij}}{l^{ij}} \; \mathrm{d} V_1^i\]</p><p>with the parameters:</p><table><tr><th style="text-align: left">size</th><th style="text-align: left">symbol</th><th style="text-align: left">unit</th></tr><tr><td style="text-align: left">bond failure</td><td style="text-align: left"><span>$d^{ij} \in \{0;1\}$</span></td><td style="text-align: left"><span>$[-]$</span></td></tr></table><p class="math-container">\[    \boldsymbol{b}_{2}^{\mathrm{int}, \, i} = 
+2 \, C_2 \int_{\mathcal{H}_2^i} d^{ijk} \left( \frac{a^{ijk}}{A^{ijk}} - 1 \right)
+\frac{\boldsymbol{\Delta x}^{ik} \times \boldsymbol{a}^{ijk}}{a^{ijk}} \; \mathrm{d} V_2^i \; .\]</p><table><tr><th style="text-align: left">size</th><th style="text-align: left">symbol</th><th style="text-align: left">unit</th></tr><tr><td style="text-align: left">material constant</td><td style="text-align: left"><span>$C_2$</span></td><td style="text-align: left"><span>$[\frac{\mathrm{kg}}{\mathrm{m}^9\mathrm{s}^2}]$</span></td></tr><tr><td style="text-align: left">(area element) failure</td><td style="text-align: left"><span>$d^{ijk} \in \{0;1\}$</span></td><td style="text-align: left"><span>$[-]$</span></td></tr></table><p class="math-container">\[\boldsymbol{b}_{3}^{\mathrm{int}, \, i} = 
+3 \, C_3 \int_{\mathcal{H}_3^i} d^{ijkl} \left( \frac{\left|{v^{ijkl}}\right|}{\left|{V^{ijkl}}\right|} - 1 \right)
+\frac{\left(\boldsymbol{\Delta x}^{ik} \times \boldsymbol{\Delta x}^{il}\right) v^{ijkl}}{\left|{v^{ijkl}}\right|} \; \mathrm{d} V_3^i\]</p><p>with the material constant <span>$C_3$</span> is used.</p><table><tr><th style="text-align: left">size</th><th style="text-align: left">symbol</th><th style="text-align: left">unit</th></tr><tr><td style="text-align: left">material constant</td><td style="text-align: left"><span>$C_3$</span></td><td style="text-align: left"><span>$[\frac{\mathrm{kg}}{\mathrm{m}^{13}\mathrm{s}^2}]$</span></td></tr><tr><td style="text-align: left">(volume element) failure</td><td style="text-align: left"><span>$d^{ijkl} \in \{0;1\}$</span></td><td style="text-align: left"><span>$[-]$</span></td></tr></table><p class="math-container">\[\boldsymbol{b}^{\mathrm{int},i} = \boldsymbol{b}_1^{\mathrm{int},i} + \boldsymbol{b}_2^{\mathrm{int},i} + \boldsymbol{b}_3^{\mathrm{int},i} \; .\]</p></article><nav class="docs-footer"><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Thursday 25 July 2024 11:52">Thursday 25 July 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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